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1. Introduction

2. The Wave-Particle Debate

3. Achinstein's Analysis

4. Coherence

5. Coherence and the Wave-Particle Debate

6. Independent warrant

7. A Critique of the Probabilistic Approach

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1. Introduction

We challenge Achinstein's conclusions using a precise characterization of coherence that incorporates many of Whewell's insights. We show that this characterization can model the reasoning of the wave theorists in the mid-nineteenth century, thereby explaining the acceptance of the wave theory over the particle theory. We conclude with a critical comparison of the probabilistic and coherence approaches to modeling this scientific revolution, arguing that the coherence account is more computationally tractable and psychologically realistic.

The characterization of coherence that we apply to the wave-particle debate is an abstraction from connectionist models of human cognition that view thinking in terms of parallel constraint satisfaction. Coherence in a set of representations is a matter of maximizing the satisfaction of positive and negative constraints among the representations. For theory choice, the representations are propositions representing hypotheses and evidence. Positive constraints are established by explanatory relations between propositions, and negative constraints are established by relations of contradiction and competition. Algorithms are available for maximizing coherence construed as constraint satisfaction, and we will describe a computer simulation of how explanatory coherence supports the nineteenth-century acceptance of the wave theory of light over the particle theory.

2. The Wave-Particle Debate

It is allowed on all sides, that light either consists in the emission of very minute particles from luminous substances, which are actually projected, and continue to move, with the velocity commonly attributed to light, or in the excitation of an undulatory motion, analogous to that which constitutes sounds, in a highly elastic medium pervading the universe. (1)

In this section we briefly outline the debate which preceded this important scientific revolution and attempt to characterize both the strengths and weaknesses of the particle and wave positions. Indisputably, the most influential voice on the side of the particle theorists was that of Newton. He proposed the particle theory as a rhetorical question (Newton, 1730, p. 370) :

Qu. 29. Are not the Rays of Light very small Bodies emitted from shining Substances?

Qu. 28. Are not all Hypotheses erroneous, in which Light is supposed to consist in Pression or Motion, propagated through a fluid Medium?

Though the wave theory of light had been supported in the early seventeenth century by Descartes (1596-1650) and the Dutch mathematician Huygens (1629-1695), it lost general acceptance upon publication of Newton's particle theory in Opticks. The resuscitation of the wave theory is often attributed to the writings of Thomas Young, which did not appear until the early years of the nineteenth century (Fresnel, 1868; Cantor, 1983; Buchwald, 1989; Achinstein, 1991) . Drawing on the writings of contemporaries such as Fresnel, Lloyd, and Herschel, Young noted a number of severe problems with the particle theory (Achinstein, 1991) .

One of the most convincing reasons offered to dismiss the particle theory of light was derived from a series of experiments and analyses performed by Fresnel. (2) These are the results Young relies upon when he notes (Young, 1807, p. 458, italics added) :

...when a portion of light is admitted through an aperture, and spreads itself in a slight degree in every direction. In this case, it is maintained by Newton that the margin of the aperture possesses an attractive force, (3) which is capable of inflecting the rays: but there is some improbability in supposing that bodies of different forms and of various refractive powers should possess an equal force of inflection, as they appear to do in the production of these effects...which is a condition not easily reconciled with other phenomena.

Furthermore, Young notes the difficulty that a particle theory of light has in explaining the observed uniform velocity of light (Young, 1855, p. 79) :

How happens it that, whether the projecting force is the slightest transmission of electricity, the friction of two pebbles, the lowest degree of visible ignition, the white heat of a wind furnace, or the intense heat of the sun itself, these wonderful corpuscles are always propelled with one uniform velocity?

Nevertheless, the alternative to the particle theory, the undulatory or wave theory, had its own difficulties (Chen, 1988) . One of the most commonly cited criticisms against the wave theory is one that was forwarded by Newton in his Opticks: "If it [light] consisted in Pression or Motion, propagated either in an instant or in time, it would bend into the Shadow" (Newton, 1730, p. 362) - there had been no such observed diffraction.

However, the more serious difficulties with the particle theory, and new evidence, including Young's own famous double-slit experiment, eventually convinced the scientific community that the wave theory was the better of the two. Table 1, taken from Reverend B. Powell's Remarks on Mr. Barton's Reply (1833), was produced by a wave theorist though it was intended as "a synoptic sketch, which I believe to be perfectly impartial; indeed, I have given every advantage to the corpuscular theory" (Powell, 1833, p. 416) .

Of course, neither the wave theory nor the particle theory were, or claimed to be, the perfect theory of light (e.g. neither explained complex coloured fringes of apertures and shadows). As new experimental cases arose, they were used to confirm or challenge the theories (Powell, 1833, p. 414) :

Mr. Barton has brought forward a new experimental case, - and the science of theoretical optics is under great obligations to him for doing so, - a case to which neither the undulatory nor any other theory ... has as yet been applied. It remains to be seen how they may apply; and this case will form a further test of the powers of either theory when formulae applying to this case shall have been investigated.

Table 1 - Synoptic sketch of the successes and difficulties of the particle and wave theories (Powell, 1833, p 416-417) .

3. Achinstein's Analysis

1. Assume that either theory T1 or theory T2 is correct, and give grounds for such an assumption.

2. Show how T1 and T2 explain various observed phenomena.

3. Show that T2 in explaining one or more of these phenomena introduces improbable hypotheses, whereas T1 does not.

4. Conclude that T1 is probably true.

Achinstein goes into great detail explaining how a wave theorist would argue for the low probability of the particle theory. He relies on the probability calculus to explicate the process wave theorists supposedly used to prove their rivals improbable (Achinstein, 1991, pp. 85-90) . He concluded that the wave theorists' strategies "can be analyzed in inductive and probabilistic terms" but not by relying on the concept of coherence (Achinstein, 1991, p. 111, 133) .

In criticizing coherentist analyses of the conceptual change which took place, Achinstein discusses the debate between John Stuart Mill and William Whewell concerning the verification of hypotheses in science (Achinstein, 1991, p. 117-148) . Achinstein characterizes the Whewellian, or coherentist, position as insisting that (Achinstein, 1991, p. 118) :

...an hypothesis explain [not only] known phenomena but that it explain and/or predict new ones as well particularly ones different in kind from those it was initially designed to explain ("consilience"). Recognizing that hypotheses being considered are usually additions to larger systems, Whewell also imposes a requirement that such additions render the system more coherent.

Nonetheless, Achinstein has noticed important difficulties with the coherentist position. The two which are most central to his dismissal of the coherentist position are that "coherence" is a poorly defined concept (Achinstein, 1991, p. 129) :

Coherence is a vague notion, for which Whewell (like most others who invoke this idea) offers no definition.

These three formulations of the method (of coherence), although by no means identical, have in common the basic idea that the fact that an hypothesis if true would correctly explain phenomena counts as some reason for believing that hypothesis....there is no requirement that the hypothesis in question, or any subsequent one, be inductively inferable from any observations. More generally, there is no requirement that there be any independent warrant for the hypotheses introduced.

4. Coherence

1. Elements are representations such as concepts, propositions, parts of images, goals, actions, and so on.

2. Elements can cohere (fit together) or incohere (resist fitting together). Coherence relations include explanation, deduction, facilitation, association, and so on. Incoherence relations include inconsistency, incompatibility, and negative association.

3. If two elements cohere, there is a positive constraint between them. If two elements incohere, there is a negative constraint between them.

4. Elements are to be divided into ones that are accepted and ones that are rejected.

5. A positive constraint between two elements can be satisfied either by accepting both of the elements or by rejecting both of the elements.

6. A negative constraint between two elements can be satisfied only by accepting one element and rejecting the other.

7. The coherence problem consists of dividing a set of elements into accepted and rejected sets in a way that satisfies the most constraints.

More precisely, consider a set E of elements which may be propositions or other representations. Two members of E, e1 and e2, may cohere with each other because of some relation between them, or they may resist cohering with each other because of some other relation. We need to understand how to make E into as coherent a whole as possible by taking into account the coherence and incoherence relations that hold between pairs of members of E. To do this, we can partition E into two disjoint subsets, A and R, where A contains the accepted elements of E, and R contains the rejected elements of E. We want to perform this partition in a way that takes into account the local coherence and incoherence relations. For example, if E is a set of propositions and e1 explains e2, we want to ensure that if e1 is accepted into A then so is e2. On the other hand, if e1 is inconsistent with e3, we want to ensure that if e1 is accepted into A, then e3 is rejected into R. The relations of explanation and inconsistency provide constraints on how we decide what can be accepted and rejected.

Formally, Thagard and Verbeurgt (forthcoming) define a coherence problem as follows. Let E be a finite set of elements {ei} and C be a set of constraints on E understood as a set {(ei, ej)} of pairs of elements of E. C divides into C+, the positive constraints on E, and C-, the negative constraints on E. With each constraint is associated a number w, which is the weight (strength) of the constraint. The problem is to partition E into two sets, A and R, in a way that maximizes compliance with the following two coherence conditions:

1. if (ei, ej) is in C+, then ei is in A if and only if ej is in A.

2. if (ei, ej) is in C-, then ei is in A if and only if ej is in R.

Intuitively, if two elements are positively constrained, we want them either to be both accepted or both rejected. On the other hand, if two elements are negatively constrained, we want one to be accepted and the other rejected. Note that these two conditions are intended as desirable results, not as strict requisites of coherence: the partition is intended to maximize compliance with them, not necessarily to ensure that all the constraints are simultaneously satisfied, since simultaneous satisfaction may be impossible. The partition is coherent to the extent that A includes elements that cohere with each other while excluding ones that do not cohere with those elements. We can define the coherence of a partition of E into A and R as W, the sum of the weights of the constraints on E that satisfy the above two conditions. Coherence is maximized if there is no other partition that has greater total weight. Maximizing coherence is a computationally very difficult problem, but algorithms are available for providing good approximations (Thagard and Verbeurgt, forthcoming) .

5. Coherence and the Wave-Particle Debate

1. Symmetry: Explanatory coherence is a symmetric relation, unlike, say, conditional probability.

2. Explanation: (a) A hypothesis coheres with what it explains, which can either be evidence or another hypothesis; (b) hypotheses that together explain some other propositions cohere with each other; and (c) the more hypotheses it takes to explain something, the less the degree of coherence.

3. Analogy: Similar hypotheses that explain similar pieces of evidence cohere.

4. Data Priority: Propositions that describe the results of observation have a degree of acceptability on their own.

5. Contradiction: Contradictory propositions are incoherent with each other.

6. Competition: If P and Q both explain a proposition, and if P and Q are not explanatorily connected, then P and Q are incoherent with each other. (P and Q are explanatorily connected if one explains the other or if together they explain something.)

7. Acceptability: The acceptability of a proposition in a system of propositions depends on its coherence with them.

Application of ECHO to the wave-particle debated does not require any special assumptions about the weights on the various constraints. In running the program, we use one default weight for all positive constraints (typically .04) and one default weight for all negative constraints (typically -.06). The positive constraints are weakened when more than one hypothesis is required to make an explanation; for example, if two propositions are required to explain something, then the positive constraints are automatically halved to .02. Sensitivity analyses show that the actual values of these weights are unimportant, so long as the absolute value of the default weight on negative constraints is greater than the default weight on positive constraints, which ensures that contradictory propositions are not both accepted.

ECHO accepts the wave theory propositions rather than the particle theory propositions for several reasons that together enter into the calculation of explanatory coherence. The wave theory explains many pieces of evidence that the particle theory does not, for example, E7 "Light has a central spot". Most of the wave theory's explanations are simple, employing only the single hypothesis WH1, "Light is a wave." In contrast, the particle theory has to invoke several additional auxiliary hypotheses such as PH2, the theory of fits. Moreover, the particle theory suffers imply NE 23, "The mass and shape of slip does (should) affect dispersion, which is contradicted by the actual evidence E23. Thus ECHO takes into account constraints based on explanatory breadth, simplicity, and negative evidence that generate the conclusion that coherence is maximized by accepting the wave theory and rejecting the particle theory. On our view, theory acceptance is inference to the best explanation, where the best explanation is calculated by a judgment of explanatory coherence construed in terms of constraint satisfaction.

Our account of theory acceptance and our input to ECHO stated in the appendix do not presuppose any special theory of explanation. In the context of the wave and particle theories of light, explanation is plausibly construed as a causal relation: that light is a wave explains why it reflects because light's being a wave causes it to reflect. The nature of causality in turn can be understood in terms of physical processes that transform matter and/or energy. Explanation, however, has many aspects and construing theory choice in terms of explanatory coherence is compatible with various ways of understanding causality and explanation (Thagard, 1992b, ch. 5).

6. Independent Warrant

1. Assume that the wave and particle theories conflict by virtue of contradictions and competitions between them.

2. Explanations by the two theories establish coherence relations.

3. The particle theory introduces implausible hypotheses, that is, ones which cohere with propositions that contradict the available evidence.

4. Accept the wave theory as part of the most maximally coherent partition of the set of relevant propositions.

Achinstein takes the concept of independent warrant from Mill and Herschel who claim, in contrast to Whewell, that explanatory power, consilience, simplicity, and coherence are not enough to justify acceptance. In addition to a hypothesis explaining observed phenomena, there should be data providing some independent inductive support for the hypothesis (Achinstein, 1991, p. 134) . Achinstein employs an "eliminative strategy" to produce an independent warrant, and claims that such a method is the one employed by wave theorists (Achinstein, 1991, p. 137) . He claims that wave theorists could argue inductively that light was either a wave or a particle, then justify acceptance of the wave hypothesis by elimination of the particle hypothesis.

But Achinstein does not show how such an argument was developed by the wave theorists, or how it might be developed. In principle, we can imagine that light might be something quite unlike a wave or a particle, perhaps an emanation from the incorporeal minds of divine beings. In the scientific context of the nineteenth century, when the wave and particle theories were the only live competitors, it was reasonable to base theory choice on a comparative assessment of them alone. Achinstein provides no textual evidence to support the claim that the wave theorists were using an inductively supported argument for the exclusivity of wave and particle approaches to justify their acceptance of the wave theory.

The reasonableness of focusing just on wave and particle theories does not derive from the existence of independent inductive support for their being the only possibilities, but from their being the only theories available with any explanatory record. Our coherence model of the debate handles this aspect adequately by basing the acceptance of the wave theory and the simultaneous defeat of the particle theory on the former's greater explanatory breadth, simplicity, and avoidance of negative evidence. Historically and normatively, that was all that the wave theorists needed. An account of explanatory coherence that takes into account competition between theories and negative evidence (negative constraints) and the explanatory relations among propositions (positive constraints) can adequately model the wave theorists position in a way that supports Whewell's view: Mill's call for independent warrant neither describes nor prescribes the nature of scientific deliberation.

7. A Critique of the Probabilistic Approach

The objection to this explanation offered by nineteenth-century wave theorists is not that it introduces ideas foreign to the particle theory, ones that render the set of theoretical assumptions "incoherent," but that it introduces hypotheses that are improbable, given observations made in this case and others.

Although the invention of plausible hypotheses, independent of any connexion with experimental observations, can be of very little use in the promotion of natural knowledge; yet the discovery of simple and uniform principles, by which a great number of apparently heterogeneous phenomena are reduced to coherent and universal laws, must ever be allowed to be of considerable importance towards the improvement of human intellect.

Our coherence account has two major advantages over Achinstein's probabilistic account: computational tractability and psychological plausibility. Connectionist coherence models like ECHO provide a computationally efficient way of dealing with large numbers of elements (propositions). (5) In contrast, Achinstein has no working model of how probabilistic reasoning can produce the acceptance of the wave theory. Producing such a probabilistic model is a difficult task. In order to include the thirty-one propositions that are used in the ECHO model, a probabilist would need 2E31 (2,147,483,648) probabilities for a full joint distribution, more than two billion probabilities.

Hence the probabilist must rely on simplifications. One such method of simplifying probabilistic networks (6) has been developed by Pearl (1988). The resulting networks (Pearl networks) are directed, acyclic graphs whose nodes are multi-valued variables and whose directed edges indicate a causal relationship. By localizing probability calculations through independence assumptions, and introducing methods for translating multiply connected networks to singly connected ones, Pearl has been able to greatly simplify the solution of probabilistic networks (Pearl, 1988) .

In a comparison of probabilistic networks to ECHO, Thagard (forthcoming) formulates a method for translating any problem captured by ECHO into a Pearl probabilistic network. However, there is a high price to be paid for such a translation: there is no straightforward method for assigning conditional probabilities in a Pearl network to give the correct results; the size of the Pearl network becomes very large, and the independence assumptions which must be made may be poor approximations (Thagard, forthcoming) . Nevertheless, one can create a Pearl network that greatly reduces the number of probabilities necessary to model a specific situation. For the 31 propositions in the ECHO simulation of the wave-particle debate, the probabilist can get by with 184 conditional probabilities, a massive improvement over 2 billion. However, this reduction is only possible if "one knows the conditional probability distribution of each propositional variable given its parents" (Neapolitain, 1990, p. 164) .

Probabilists know that finding these numbers is a difficult task. Though humans are quite proficient at creating a causal network, and even estimating the probability an event will occur, it is much more difficult for them to determine the values in the conditional distributions of a node given its parents (Neapolitain, 1990, p.181) . For example, one would have to estimate the probability that the theory of fits is correct, given that light exhibits colours of thin plates (see the appendix). The probability can not be assumed to be 1, as there are other theories which may account for this property of light. It is also unclear how a probability estimation will change with the introduction of new evidence, or even which evidence relates to which theories. How then, does one arrive at a reasonable probability value under such uncertain conditions? This question is not resolved by Achinstein's analysis.

Furthermore, there are important restrictions to creating a probabilist model. In creating a probabilist net one can not simply specify a (directed acyclic) network and the conditional probability distributions of every variable given its parents. First, one must show that the specified conditional distributions determine a joint probability distribution. Second, it must be shown that the specified conditional probabilities are indeed the conditional probabilities. Finally, the joint distribution and the graph must be shown to satisfy the conditional independence assumptions of the probabilistic network (Neapolitain, 1990, p. 164) . The only one of these tasks that is necessary for the coherentist corresponds to the creation of the probabilist network, the simplest of all the probabilist tasks. Moreover, the strong independence assumptions necessary to reduce the probabilistic network's complexity tractable are unrealistic. If one examines the particle-wave debate, it is quite clear that a number of hypotheses are dependent on each other, particularly in the case of the particle hypotheses listed in the appendix. We issue the following challenge to proponents of probabilistic accounts of scientific reasoning: develop a historically plausible and tractable computational model of how probabilities can be used to justify the acceptance of the wave theory. Our ECHO simulation shows that the coherence account already satisfies the criterion of computational tractability.

Our coherence account operates in the tradition of naturalistic epistemology, since connectionist models of mind have received much empirical support in recent years, and ECHO and related models of analogy and concept application have been used to explain and predict the results of many psychological experiments (on empirical applications of ECHO, see Byrne, 1995; Read and Marcus-Newhall, 1993; Schank and Ranney, 1992; for the general approach, see Thagard, in press, ch. 7). In contrast, there is abundant psychological evidence that reasoning with probabilities is not a natural part of human cognition (Kahneman, Tversky, and Slovic, 1992). Hence our coherence account of scientific change is more psychologically plausible than probabilistic accounts as well as more computationally tractable.

In sum, we have used a computationally implemented theory of explanatory coherence to give a historically accurate and thorough analysis of the acceptance of the wave theory of light. This analysis should not be judged in terms of how well it approximates and motivates a probabilistic account of the wave-particle debate, since there is no reason to believe that debate was fundamentally probabilistic. Probability theory is wonderful in statistical contexts where samples from populations allow us to speak of frequencies and propensities, but there is no reason why belief revision in human scientists needs to be interpreted probabilistically. Just as nineteenth-century wave theorists and particle theorists competed to explain the nature of light, so coherence theorists and probability theorists compete to explain the nature of scientific change. We have provided in this paper a coherence account of the acceptance of the wave theory of light that is at least as historically accurate as Achinstein's probabilist account and that is superior in computational tractability and psychological plausibility.

Appendix: ECHO simulation

;; Wave hypotheses

(proposition 'WH1 "Light is a wave.")

(proposition 'WH2 "Light has transverse vibrations.")

;; Particle hypotheses

(proposition 'PH1 "Light is a series of particles emitted from a source.")

(proposition 'PH2 "Theory of fits.")

(proposition 'PH3 "Theory of movable polarization.")

(proposition 'PH4 "Theory of inflection.")

(proposition 'PH5 "Theory of polarity.")

;; Empirical evidence

(proposition 'E1 "Light reflects.")

(proposition 'E2 "Light reflects at the boundary of transparent medium.")

(proposition 'E3 "Light refracts, it is homogeneous.")

(proposition 'E4 "Light disperses.")

(proposition 'E5 "Light can be absorbed.")

(proposition 'E6 "Light exhibits colours of thin plates.")

(proposition 'E7 "Light has a central spot (around circular objects).")

(proposition 'E8 "Light exhibits Airy's modification.")

(proposition 'E9 "Light exhibits thick plates.")

(proposition 'E10 "Light has coloured fringes of apertures and shadows in simple cases.")

(proposition 'E11 "Light has coloured fringes in more complex cases.")

(proposition 'E12 "Light stripes if it is mixed.")

(proposition 'E13 "Light shifts if a plate is interposed.")

(proposition 'E14 "Light has colours through gratings.")

(proposition 'E15 "Light has double refraction.")

(proposition 'E16 "Light can be polarized.")

(proposition 'E17 "Light has a connexion with double refraction.")

(proposition 'E18 "Light exhibits the law of tangents.")

(proposition 'E19 "Light when polarized has interference patterns.")

(proposition 'E20 "Light has polarization rings.")

(proposition 'E21 "Light can be circularly and elliptically polarized.")

(proposition 'E22 "Light has conical refraction.")

(proposition 'E23 "The mass and shape of slit does not affect dispersion.")

(proposition 'NE23 "The mass and shape of slit does (should) affect dispersion.")

(contradict 'E23 'NE23)

(data '(E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15 E16 E17 E18 E19 E20 E21 E22 E23))

The program takes these inputs and creates a constraint network in which coherence relations are implemented as positive constraints represented by excitatory links and incoherence relations are implemented as negative constraints implemented by inhibitiory links. A simple connectionist algorithm spreads activation among the propositions until some are accepted (activation greater than 0) and others are rejected (activation less than 0). See Thagard (1992) for mathematical details. Table 2 displays the final activations of the main hypotheses in the wave/particle dispute. We have done computational experiments that show that explanatory breadth, simplicity, and negative evidence all contribute to the superiority of the wave hypotheses over the particle hypotheses.

Hypotheses	Final Activation
WH1	0.94
WH2	0.67
PH1	-0.85
PH2	-0.57
PH3	-0.57
PH4	-0.66
PH5	-0.63

Table 2 - Activation levels of hypotheses in coherentist analysis of the wave particle debate.

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